\usepackage{graphicx}
\usepackage{color}
\usepackage{latexsym}
\usepackage{balance}
\usepackage{subfigure}
\usepackage{setspace}  
\usepackage{amssymb}
\usepackage{amsmath} 
\usepackage{tikz} 
\usepackage{pgfplots}
\pgfplotsset{every x tick label/.append style={font=\tiny, yshift=0.5ex}}
\usepackage[justification=centering]{caption}

\newcommand{\todo}[2]{{\underline{\textsf{#1}}:} {\color{red}{$\spadesuit$#2$\spadesuit$}}}

\def\PT#1{\hbox{\textrm{pt}$_{#1}$}}
\def\Nc#1#2{\hbox{\texttt{#1$_{#2}$}}}
\def\Nl#1#2{\hbox{$\bullet$\texttt{#1$_{#2}$}}}
\def\Nr#1#2{\hbox{\texttt{#1$_{#2}$}$\bullet$}}
\def\Np#1#2{\hbox{$\bullet$\texttt{#1$_{#2}$}$\bullet$}}
\def\modify#1#2#3{{\underline{\sf{#2}}:} {\color{blue}{#1}} {\color{green}{\mbox{$\Rightarrow$}}} {\color{red}{#3}}}
\def\hmark#1{\color{blue}({#1}) \color{black}}
\def\ST#1{\hbox{\textrm{st}$_{#1}$}}
\def\pred#1#2{#1\to\{#2\}}
%\def\INDEXSET#1{\mathit{#1}_{[0,P)}}
\def\INDEXSET#1{\mathit{#1}_{[P]}}
\def\INDEXSETL#1{\mathit{#1}_p}
%\newtheorem{definition}{Definition}
%\newtheorem{property}{Property}

